My questions come from the paper Logarithmic Sobolev inequalities for some nonlinear PDE’s written by F. Malrieu (May 2001). The basic set-up is that we have a $N$-particle system $(X^{i,N}_t)_{1\leq i\leq N}$ satisfying a coupled system of Mckean-Vlasov SDEs, with the law of $(X^{i,N}_0)$ being independent and identically distributed (see Equation (3) in their paper). The (first) uniform in time propagation of chaos proved in their paper is Theorem 3.3 (see Equation (9)), i.e., $$\sup\limits_{t \in \mathbb R} \mathbb{E}\left(\left|X^{1,N}_t - \overline{X}^1_t\right|^2\right) \leq \frac{K}{N}\tag{a} $$ for some fixed $K > 0$, where $(\overline{X}^1_t)$ is the solution of a nonlinear Mckean-Vlasov SDE (see Equation (2)) with the same Brownian motion as $X_t^{1,N}$ and whose law $u_t(x)$ satisfies the Mckean-Vlasov PDE (see Equation (1)). Then later in Proposition 3.13 the author provided another (uniform in time) propagation of chaos result of a somewhat different flavor: $$\text{for all}~~ k \leq N, ~~\sup\limits_{t \in \mathbb R} \|u^{(k,N)}_t - u^{\otimes k}_t \|_1 \leq K\sqrt{\frac{k}{N}}, \tag{b} $$ in which $u^{(k,N)}_t$ is the marginal law of the first $k$-particles among the $N$-particle system and $u_t$ is the solution of the Mckean-Vlasov PDE (see Equation (1)). Although I have no issue in understanding the proofs of both results, the statement before the statement of Proposition 3.13 really confuses me, it said "We are now able to improve the propagation of (chaos) result (9) [(a) above]." However, I have no clue as to why (b) is an improvement over (a), and I have not figured out how to deduce (a) from (b). Thank you very much for any help!
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$\begingroup$ According to my reading, Malrieu’s work is really about the nonlinear McKean Vlasov PDE, and in particular, proving the main result on its convergence to equilibrium Theorem 1.3. Therefore, everything else (including the corresponding SDE, the mean-field particle system/PDE, etc.) has only instrumental value towards this main goal. From this viewpoint, Prop 3.13 is indeed an improvement over Theorem 3.3 because it is closer to the main focus of the paper. $\endgroup$– Nawaf Bou-RabeeCommented Jun 2, 2022 at 13:00
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$\begingroup$ @NawafBou-Rabee Hello Professor, "Prop 3.13 is indeed an improvement over Theorem 3.3 because it is closer to the main focus of the paper" sounds like a "philosophical" justification of Malrieu’s claim instead of "mathematical". From my understanding, to say that (b) is an improvement over (a) you need to be able to "derive" (a) from (b)... $\endgroup$– Fei CaoCommented Jun 2, 2022 at 15:33
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$\begingroup$ It is clear from the proof of (b) that they only use (a) to prove (b) and certainly not the converse statement. Therefore, I think improvement here really means something else. $\endgroup$– Nawaf Bou-RabeeCommented Jun 2, 2022 at 15:43
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$\begingroup$ @NawafBou-Rabee I agree with the fact that the author used (a) to prove (b), but still that does not mean (b) is "stronger" than (a) (in whatever rigorous sense). But you are right, the so-called "improvement" probably means something else... $\endgroup$– Fei CaoCommented Jun 2, 2022 at 16:14
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$\begingroup$ This is why I mainly focus on theorems in math papers, since they are rigorous, while the words in between are often not so rigorous. $\endgroup$– Nawaf Bou-RabeeCommented Jun 2, 2022 at 16:23
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